Page 340 - Matrix Analysis & Applied Linear Algebra

336 Chapter 5 Norms, Inner Products, and Orthogonality
 
1+ i 1+ i
√ √
 3 6 
5.6.2. Is   a unitary matrix?
 i −2 i 
√ √
3 6
5.6.3. (a) How many 3 × 3 matrices are both diagonal and orthogonal?
(b) How many n × n matrices are both diagonal and orthogonal?
(c) How many n × n matrices are both diagonal and unitary?

5.6.4. (a) Under what conditions on the real numbers α and β will

α + β β − α
P =
α − β β + α
be an orthogonal matrix?
(b) Under what conditions on the real numbers α and β will

 
0 α 0 iβ
 α 0 iβ 0 
0 iβ 0 α
U =  
iβ 0 α 0
be a unitary matrix?

5.6.5. Let U and V be two n × n unitary (orthogonal) matrices.
(a) Explain why the product UV must be unitary (orthogonal).
(b) Explain why the sum U+V need not be unitary (orthogonal).
0
(c) Explain why U n×n must be unitary (orthogonal).
0 V m×m
5.6.6. Cayley Transformation. Prove, as Cayley did in 1846, that if A is
skew hermitian (or real skew symmetric), then
U =(I − A)(I + A) −1 =(I + A) −1 (I − A)

is unitary (orthogonal) by first showing that (I + A) −1 exists for skew-
hermitian matrices, and (I − A)(I + A) −1 =(I + A) −1 (I − A) (recall
Exercise 3.7.6). Note: There is a more direct approach, but it requires
the diagonalization theorem for normal matrices—see Exercise 7.5.5.

5.6.7. Suppose that R and S are elementary reflectors.
I 0
(a) Is an elementary reflector?
0 R
R 0
(b) Is an elementary reflector?
0 S